Automorphisms of Fano threefolds of rank 2 and degree 28

We describe the automorphism groups of smooth Fano threefolds of rank 2 and degree 28 in the cases where they are finite.


Introduction
A smooth Fano threefold of Picard rank 2 and degree 28 is the blow-up of a smooth quadric threefold Q ⊂ P 4 in a smooth rational quartic curve C 4 ⊂ Q.The family of all such threefolds is denoted №2.21.Let π : X → Q be such a threefold.Then the action of Aut(Q, C 4 ) on Q lifts to an action on X, so that we may identify it with a subgroup of Aut(X).
By a result of Cheltsov-Przyjalkowski-Shramov ([CPS21, Lemma 9.2]), we have that either Aut(X) is finite, or Aut(X) ∼ = Aut(Q, C 4 )×Z 2 , where upto isomorphism Aut(Q, C 4 ) is described as follows: (1) There is a unique smooth threefold in №2.21 such that Aut(Q, C 4 ) ∼ = PGL 2 (C), (2) There is a one-dimensional family of non-isomorphic smooth threefolds in №2.21 such that Aut(Q, C 4 ) ∼ = G m ⋊ Z 2 , (3) There is a unique smooth threefold in №2.21 such that Aut(Q, C 4 ) ∼ = G a ⋊ Z 2 .The goal of this paper is to describe Aut(X) when it is finite, where X is a smooth threefold in family №2.21.Our main result is the following: Theorem 1.1.Let X be a smooth Fano threefold of rank 2 and degree 28.Then Aut(X) ∼ = Aut(Q, C 4 ) × Z 2 .Furthermore, if Aut(Q, C 4 ) is finite then it is isomorphic to Z 2 × Z 2 , Z 2 or 0.
We prove this theorem in two parts: Theorem 2.1 and Theorem 3.1.
Remark 1.2.The factor of Z 2 appearing in the factorisation Aut(X) ∼ = Aut(Q, C 4 ) × Z 2 is generated by an involution g, which may be described as follows: Let d denote the restriction to Q of the linear system of quadric hypersurfaces in P 4 which contain C 4 , and let φ : Q P 4 be the corresponding rational map.The image of φ is a smooth quadric threefold, and φ contracts the intersection of the secant variety of C 4 with Q, V , onto a smooth rational curve C ′ 4 ⊂ Q ′ .The base locus of φ is equal to C 4 , so that there is a birational morphism π ′ : X → Q ′ , where X is the blow-up of Q along C 4 .This morphism contracts the strict transform, E ′ , of V onto the curve C ′ 4 .Thus, there is a commutative diagram In [CPS21], it is shown that in cases (1) and (2) of the above classification, there exists a basis of d such that Q ′ = Q and C ′ 4 = C 4 , so that φ lifts to an involution g ∈ Aut(X).We will show in Theorem 3.1 that this is always the case.
We can explicitly describe the threefolds appearing in [CPS21, Lemma 9.2].Let us fix some notation.Observe that after a projective transformation C 4 is the image of the Veronese embedding of P 1 in P 4 : The space of global sections of I C 4 (2) is generated by the following quadratic forms: , where I C 4 is the ideal sheaf of C 4 in P 4 , and x 0 , x 1 , x 2 , x 3 , x 4 are homogeneous coordinates on P 4 .Observe that the standard PGL 2 (C)-action on C 4 lifts to an action on P 4 such that C 4 is invariant.We fix the following subgroups of PGL 2 (C): G a , consisting of matrices 1 t 0 1 for every t ∈ G a .
Now we can describe Aut(X) for the threefolds listed before: Example 1.3.([Ara+23, Section 5.9]).Let Q be the quadric given by the equation and Z 2 -invariant, and conversely any smooth quadric admitting a faithful G m -action is isomorphic, via an element of PGL 2 (C), to a quadric given by an equation of this form.Moreover, we have the following: 2 .The involution g described before is given by: See [Ara+23,Remark 5.52] for an explanation of why τ Example 1.4.Suppose that the quadric Q is given by the equation Then Q is G a -invariant and Z 2 -invariant, and Aut(Q, C 4 ) ∼ = G a ⋊ Z 2 .We will prove in case (2) of Theorem 3.1 that the blow-up of Q in C 4 admits an action of the involution g.Remark 1.5.Recall that for a finite subgroup G ⊂ Aut(Y ), a variety Y is called G-Fano if it has terminal singularities, −K Y is ample and Cl(Y ) G is rank 1.It is proven in [KPS18] that the Hilbert scheme of conics on a smooth threefold X from the family №2.21 is isomorphic to P 1 × P 1 , with the degenerate conics being parameterised by a smooth curve then this curve must be invariant upon swapping the two factors of P 1 .
An informal conjecture of Y. Prokhorov is that the invariance of this curve is a sufficient condition for X to be G-Fano.It is proven in [Che+23] that every smooth curve in P 1 × P 1 of bidegree (2, 2) is invariant.As a corollary to Theorem 1.1, we have that every smooth threefold X in the family №2.21 is G-Fano, so that Prokhorov's informal conjecture is true.For a detailed discussion of G-Fano threefolds, see [Sar22].
Remark 1.6.Smooth threefolds in the family №2.21 are parametrised by P 5 \ ∆, where ∆ ⊂ P 5 is the discriminant locus of singular quadrics.The group PGL 2 (C) acts on this space, and it follows from Theorem 1.1 that any two threefolds in №2.21 are isomorphic if and only if their corresponding points in the parameter space P 5 \ ∆ lie in the same PGL 2 (C)-orbit.Moreover, the moduli space of smooth GIT-polystable threefolds in №2.21 is given by the GIT quotient (P 5 \ ∆)//PGL 2 (C).
Acknowledgements.I am grateful to I. Cheltsov for introducing me to this topic, and for his attentive guidance, support and insight with regards to the creation of this document.I would also like to thank I. Dolgachev for his useful observation in the proof of case (1) in Theorem 3.1.

Computation of Aut
The first half of proving Theorem 1.1 is the computation of Aut(Q, C 4 ), which we will do in this section.The result we will prove is: The following lemma will be useful: Lemma 2.2.Let Q ⊂ P 4 be any quadric hypersurface containing the curve C 4 .Suppose that Aut(Q, C 4 ) is finite, and contains an element of finite order n > 2. Then Q is singular.
Proof of Theorem 2.1.We may assume that µ = 0 in case (1), and λ = 0 in case (2), as otherwise the threefolds are isomorphic to those which are described in Example 1.3 and Example 1.4.Then Aut(Q, C 4 ) is finite, and since Aut(Q, C 4 ) is isomorphic to a subgroup of PGL 2 (C), it must be isomorphic to one of the following groups: where S n (resp.A n ) is the symmetric (resp.alternating) group on n letters.Then by Lemma 2.2 the only possibilities are that Aut(Q, Suppose Q is in case (1).Then Q admits an action of Z 2 × Z 2 , generated by g 1 , g 2 ∈ PGL 2 (C), which are given by: Suppose that Q is in case (2).Then Q admits an action of the group Z 2 , generated by the element g 1 .Suppose that Aut(Q, C 4 ) ∼ = Z 2 × Z 2 , and let g ∈ Aut(Q, C 4 ) be a non-trivial element distinct from g 1 .Considering the standard action of PGL 2 (C) on P 1 , observe that g 1 fixes the points [0 : 1] and [1 : 0], and since gg 1 = g 1 g, we see that g must swap these points.Since g has order 2, it must be equal to either g 2 or g 1 g 2 .The threefold Q is not invariant under either of these.
Finally suppose that Q is in case (3), and suppose that Aut(Q, C 4 ) is non-trivial.Then it contains an element, g, of order 2. Since g fixes two distinct points of P 1 , it must be equal to g 2 , g 1 g 2 , or be given by a matrix of the form 1 a b −1 , for some a, b ∈ C such that ab = −1.
One checks that g 2 nor g 1 g 2 leave Q invariant, and if g is given by a matrix of the above form then g(Q) is given by the equation: Clearly g(Q) = Q, so that Aut(Q, C 4 ) has to be trivial.

Existence of a birational involution of Q
The second half of proving Theorem 1.1 is the assertion that Aut(X) ∼ = Aut(Q, C 4 )×Z 2 , which we will do in this section.The result is: Theorem 3.1.Let X be a smooth Fano threefold in family №2.21.Then there exists an involution g ∈ Aut(X) such that Aut(X) ∼ = Aut(Q, C 4 ) × g .
Proof.We proceed case-by-case, according to the classification in Theorem 2.3.
Case (1): Q is given by µ(f 0 + f 4 ) + λf 2 + f 5 = 0. Observe that the linear system of quadrics which contain C 4 is 5-dimensional, so it is more natural to express members of family №2.21 in terms of fourfolds.Let us show how to do this.Fix the Veronese surface S 4 ⊂ P 5 given by the embedding: The space of global sections of I S 4 (2) is generated by the quadratic forms: where x 0 , x 1 , x 2 , x 3 , x 4 , x 5 are homogeneous coordinates on P 5 .Consider the following rational map: I claim that φ is a birational involution.The following observation is due to I. Dolgachev: we can identify P 5 with the space of symmetric 3 ×3 matrices, upto scaling.Then under this identification, the rational map above is: But this is the same map as taking a matrix M to its adjoint adj(M).Thus it follows from the relation adj(adj(A)) = det(A) n−2 A for any n × n matrix A that φ is a birational involution.
Let σ : P 5 → P 5 be the blow-up of P 5 in S 4 , and let E be the exceptional divisor.Observe that for general divisors Since φ has base locus equal to S 4 , it lifts to a biregular involution g ∈ Aut( P 5 ) which swaps the linear systems | H| and | Q|.Thus, the intersection H ∩ g( H) is g -invariant, for any H.We will now show that every smooth element X of №2.21 which is in case (1) of Theorem 2.3 is isomorphic to a subvariety of P 5 of the form H ∩ g( H), for some hyperplane H ⊂ P 5 , and therefore possesses an involution not coming from Aut(Q, C 4 ).So fix such a threefold X.Then the quadric Q is given by the equation Now consider the following hypersurfaces in P 5 : We have that the intersections H ∩ S 4 and Q 2 ∩ S 4 are smooth, so that the intersection of their strict transforms, H ∩ Q 2 ⊂ P 5 , is a smooth member of №2.21.Moreover, Consider the projective transformation ψ : P 5 → P 5 given by the matrix .
Then the intersection of the fourfolds is given as a subvariety of P4 by Equation 3.2, which defines X.
It remains to show that the birational involution g commutes with the action of Aut(Q, C 4 ).
Consider the subgroup G ⊂ PGL 6 (C) generated by the commuting involutions Then α and β commute with the birational involution described previously: The hypersurfaces H and Q 2 are G-invariant.Moreover S 4 is G-invariant, so that G is isomorphic to a subgroup of Aut(Q, C 4 ).Thus since Aut(Q, C 4 ) ∼ = Z 2 ×Z 2 by Theorem 2.1, we have that Aut(Q, C 4 ) ∼ = G.So we see that Aut(Q, C 4 ) commutes with the involution g.
For the remaining cases, we will compute bases for the linear system d of quadrics sections of Q ⊂ P 4 containing the curve C 4 such that the corresponding rational map is an involution, and commutes with the action of Aut(Q, C 4 ).
Case (2): Q is given by f 0 + λf 2 + f 5 = 0. Let us make the substitution λ = 1 − 4s2 , for some s ∈ C \ {−1, 0, 1}.Consider the rational map: Observe that it has base locus equal to C 4 , so indeed corresponds to the linear system d.
To see that the map ι is a birational involution, consider the following rational parametrisation of Q, . This is a rational inverse to the projection Q P 3 from the point [0 : 1 : 0 : 0 : 0].Moreover, it is an isomorphism between the open subsets P 3 \ Π and Q \ V , where Π ⊂ P 3 is the plane given by x 3 = 0, and V ⊂ Q is the singular quadric surface given by the intersection of Q with the plane x 3 = 0, this latter variety being the closure of the union of lines through [0 : 1 : 0 : 0 : 0].Let Z be the curve p −1 (C 4 ), which is a quartic rational curve in P 3 .Then ι(p(P 3 \ (Π ∪ Z)) lies in Q, and since p( it follows that ι is a rational self-map of Q.To see that ι is an involution on Q \ C 4 , observe that ι • ι • p is equal to the map which is equal to the identity morphism on P 3 \ Π.Thus ι • ι is equal to the identity morphism on Q \ (V ∪ C 4 ), so that it is equal to the identity morphism on Q \ C 4 .
Case (3): Q is given by f 1 + f 5 = 0. Let σ be the rational map By replacing the map ι with σ, the map p by the rational parametrisation : x 1 : x 2 : x 3 : x 4 ],